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Abstract We show for of equal volume and that if , then (up to translation) is bounded. This establishes the sharp threshold for the quantitative stability of the Brunn–Minkowski inequality recently established by Figalli, van Hintum, and Tiba, the proof of which uses our current result. We additionally establish a similar sharp threshold for iterated sumsets.
Hintum et al. (Thu,) studied this question.
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